Fundamental Theorem of Calculus (FTC)

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 4.9 – Antiderivatives

• Section 5.3 – The Fundamental Theorem of Calculus

• Section 5.4 – Indefinite Integrals and the Net Change Theorem

Expected Educational Results

• Objective 01–01: I can state the Fundamental Theorem of Calculus.

• Objective 01–02: I can explain the meaning of the Fundamental Theorem of Calculus – Part I.

• Objective 01–03: I can explain the meaning of the Fundamental Theorem of Calculus – Part II.

• Objective 01–04: I can determine the properties of an area function using FTC-I and Calculus I.

• Objective 01–05: I can evaluate indefinite integrals using the Fundamental Theorem of Calculus – Part I.

• Objective 01–06: I can evaluate definite integrals using the Fundamental Theorem of Calculus – Part II.

• Objective 01–07: I can use the Net Change Theorem to identify the meaning of ${\int }_a^b{f}^{\prime }(x)\text{d}x$$\int_a^b{f^{\,\prime}(x)\,\text{d}x}$.

• Objective 01–08: Given velocity of an object in motion along a line, I can find the object’s displacement and the total distance traveled.

• Objective 01–09: Given acceleration of an object in motion along a line, I can find the object’s velocity, displacement, and the total distance traveled.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

FTC-II

Theorem: Fundamental Theorem of Calculus - Part II [FTC-II]

Let $f(t)$$f(t)$ be continuous on $[a,b]$$[a,b]$. Let $F(t)$$F(t)$ be any antiderivative of $f(t)$$f(t)$. Then ${\displaystyle {\int }_a^{b}f(t)dt=F(b)-F(a)}$$\displaystyle \int_a^b{f(t)\,dt}=F(b)-F(a)$.

Check Conditions of FTC-II

1. $f(t)$$f(t)$ is the black graph. Is $f(t)$$f(t)$ continuous on $[1,12]$$[1,12]$? Explain. NOTE: A function is continuous on an interval IF:

   • there are NO holes in the graph;

   • there are NO jump discontinuities in the graph; and

   • there are NO infinite discontinuities in the graph.

Activity 16

NOTE: This DESMOS activity will help you understand the Fundamental Theorem of Calculus - Part II (FTC-II). For a good proof (which is beyond the expectation for this course) of both FTC-I and FTC-II, see https://math.berkeley.edu/~peyam/Math1AFa10/Proof of the FTC.pdf.

Procedure:

Let $g(x)={\int }_1^{a}f(t)dt$$g(x)=\int_1^a{f(t)\,dt}$. In the DESMOS page,

1. Move the slider for the parameter $a$$a$ [Box 3] to a value of $a=12$$a = 12$.

2. Identify the net area [the value listed in Box 2] under $f(t)$$f(t)$ [black curve].

3. From the graph, identify the value of the antiderivative $F(x)$$F(x)$ function [green curve] when $a=12$$a=12$.

4. From the graph, identify the value of the antiderivative $F(x)$$F(x)$ function [green curve] when $a=1$$a=1$.

5. Evaluate $F(12)-F(1)$$F(12)-F(1)$ and compare to the net area under $f(t)$$f(t)$ [black curve]. What do you notice?

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Tuesday, 25 August 2020 03:53 EDT Last Modified: Monday, 30 May 2022 - 14:32 (EDT)